A185582 Decimal expansion of Sum_{m,n,p = -infinity..infinity} 4*(-1)^(m+p)/sqrt(m^2 + (2n-1/2)^2 + (2p-1/2)^2).

3, 6, 3, 4, 8, 9, 9, 0, 1, 1, 0, 4, 9, 1, 4, 8, 7, 1, 1, 3, 6, 8, 0, 4, 2, 9, 9, 2, 0, 0, 7, 8, 1, 5, 3, 2, 7, 9, 9, 0, 1, 4, 4, 3, 0, 9, 3, 5, 5, 3, 4, 0, 1, 8, 0, 6, 2, 1, 3, 0, 9, 1, 5, 2, 6, 9, 1, 2, 1, 5, 4, 8, 4, 1, 7, 8, 4, 5, 8, 8, 7, 2, 9, 1, 0, 0, 9, 3, 7, 4, 5, 0, 9, 4, 7, 6, 5, 1, 2, 5, 8, 0, 8, 9, 1

Offset: 1

Author

R. J. Mathar, Jan 31 2011

Keywords

• cons
• nonn

Comments

The defining equation (3.10i) on page 1738 of the 1975 paper has a typo (m for n).

The value in Table 4 (page 1742, column h(2s)) seems to have two digits swapped.

Examples

			3.634899011049148711368042992007815327990...
		

Links

• I. J. Zucker, Madelung constants and lattice sums for invariant cubic lattice complexes and certain tetragonal structures, J. Phys. A: Math. Gen. 8 (11) (1975) 1734, variable h(1).
• I. J. Zucker, Functional equations for poly-dimensional zeta functions and the evaluation of Madelung constants, J. Phys. A: Math. Gen. 9 (4) (1976) 499, variable h(1).

Crossrefs

Cf. A185576, A185577, A185578, A185579, A185580, A185581, A185583.

Programs

Mathematica

digits = 105; Clear[f]; f[n_, p_] := f[n, p] = (d = Sqrt[2 n^2 + (p - 1/2)^2]; (Csch[d*Pi]/d) // N[#, digits + 10] &); f[m_] := f[m] = 4 Log[1 + Sqrt[2]] + 8*Sum[f[n, p], {n, 1, m}, {p, 1, m}] // RealDigits[#, 10, digits + 10] & // First; f[0]; f[m = 10]; While[ f[m] != f[m - 10], Print[m]; m = m + 10]; f[m][[1 ;; digits]] (* Jean-François Alcover, Feb 21 2013 *)

Formula

Equals 4*log(1+sqrt(2)) + 8*Sum_{n>=1, p>=1} cosech(d*Pi)/d where d = sqrt(2*n^2 + (p-1/2)^2).

Extensions

More terms from Jean-François Alcover, Feb 21 2013